A189025 -id:A189025 - cp's OEIS frontend

A189024 Number of primes in the range (n - sqrt(n), n].

0, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 1, 1, 1, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 1, 2, 2, 3, 3, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 3, 3, 3, 3, 4, 3, 4, 4, 4, 3, 4, 3, 3, 3, 3, 2, 2, 1, 1, 1, 1, 1, 0, 0, 1

Offset: 1

T. D. Noe, Apr 15 2011

nonn

Note that the lower bound, n-sqrt(n), is excluded from the count and the upper range, n, is included. The last zero term appears to be a(126). See A189026 for special primes associated with this sequence. This sequence is related to Oppermann's conjecture that for any k > 1 there is a prime between k^2 - k and k^2.

T. D. Noe, Table of n, a(n) for n = 1..10000
Wikipedia, Oppermann's conjecture

Cf. A188817, A189025, A189027.

cnt = 0; lastLower = 0; Table[lower = Floor[n - Sqrt[n]]; If[lastLower < lower && PrimeQ[lower], cnt--]; lastLower = lower; If[PrimeQ[n], cnt++]; cnt, {n, 100}]
Table[PrimePi[n]-PrimePi[n-Sqrt[n]],{n,130}] (* Harvey P. Dale, Mar 26 2023 *)

A189026 There appear to be at least n primes in the range (x-sqrt(x), x] for all x >= a(n).

Original entry on oeis.org

127, 1367, 2531, 2539, 6007, 7457, 10061, 10847, 23531, 35797, 35801, 38557, 44497, 47111, 69767, 69809, 88321, 107687, 110419, 110431, 113723, 127217, 250673, 250681, 250687, 250703, 268487, 268493, 286381, 286393, 302563, 302567, 360947, 369821, 405199

Offset: 1

T. D. Noe, Apr 15 2011

nonn

These terms exist only if a strong form of Oppermann's conjecture that for any k>1 there is a prime between k^2-k and k^2 is true. Note that every term is prime. Sequence A189024 gives the number of primes in the range (x-sqrt(x), x]. The index of the prime a(n), that is, primepi(a(n)), is approximately (2.4*n)^2. These primes are generated in a manner similar to the Ramanujan primes (A104272).

T. D. Noe, Table of n, a(n) for n = 1..1000
Wikipedia, Oppermann's conjecture

Cf. A189025, A189027.

A189027 There appear to be at least n primes in the range (x-2*sqrt(x), x] for all x >= a(n).

Original entry on oeis.org

2, 3, 37, 139, 331, 1409, 1423, 1427, 2239, 3163, 3181, 3511, 6547, 7433, 7457, 7487, 10061, 11777, 11779, 14401, 18899, 19081, 19373, 23537, 24763, 27617, 27673, 32027, 32051, 38113, 43573, 43579, 47269, 47279, 50839, 61463, 88643, 88651, 88657, 88729

Offset: 1

T. D. Noe, Apr 15 2011

nonn

These terms exist only if a strong form of Legendre's conjecture that there is a prime between consecutive squares is true. Note that every term is prime. Sequence A189025 gives the number of primes in the range (x-2*sqrt(x), x]. The index of prime a(n), that is, primepi(a(n)), is approximately (5n)^2. These primes are generated in a manner similar to the Ramanujan primes (A104272).

T. D. Noe, Table of n, a(n) for n = 1..1000
Wikipedia, Legendre's conjecture

Cf. A189024, A189026.

A192226 Numbers n such that all integers in the interval (n-2*sqrt(sqrt(n)), n] are composite.

Original entry on oeis.org

1, 28, 36, 96, 120, 121, 122, 123, 124, 125, 126, 146, 147, 148, 189, 190, 207, 208, 209, 210, 219, 220, 221, 222, 249, 250, 292, 302, 303, 304, 305, 306, 326, 327, 328, 329, 330, 346, 477, 478, 519, 520, 533, 534, 535, 536, 537, 538, 539, 540, 630, 672

Offset: 1

Juri-Stepan Gerasimov, Jun 26 2011

nonn

a(14432) = 191913030 is probably the last term. Any further terms must be greater than 1.5 * 10^18. - Charles R Greathouse IV, Jun 30 2011

Charles R Greathouse IV, Table of n, a(n) for n = 1..14432

Cf. A188817, A189025, A008995.

Select[Range[700],NextPrime[#-2Sqrt[Sqrt[#]]]>#&] (* Harvey P. Dale, Jul 27 2011 *)

is(n)=for(k=0,sqrtnint(16*n-1,4),if(isprime(n-k), return(0))); 1 \\ Charles R Greathouse IV, Aug 26 2015

a(12)-a(13) added, a(53)-a(56) corrected by Charles R Greathouse IV, Jun 30 2011

A192231 Numbers k without prime numbers in the range (k-3*sqrt(sqrt(k)), k].

Original entry on oeis.org

1, 123, 124, 125, 126, 306, 330, 538, 539, 540, 904, 905, 906, 1147, 1148, 1149, 1150, 1346, 1347, 1348, 1349, 1350, 1351, 1352, 1353, 1354, 1355, 1356, 1357, 1358, 1359, 1360, 1689, 1690, 1691, 1692, 1971, 1972, 2200, 2201

Offset: 1

Juri-Stepan Gerasimov, Jun 26 2011

nonn

a(934) = 20831532 is probably the last term. Any further terms must be greater than 1.5 * 10^18. - Charles R Greathouse IV, Jun 27 2011

			a(2)=123 because (123-3*sqrt(sqrt(123)), 123]=(123-9,9907.., 123]=(113,0092.., 123].

Charles R Greathouse IV, Table of n, a(n) for n = 1..934

Subsequence of A192226.

Cf. A001223, A188817, A189025, A192926.

Select[Range[2000], PrimePi[# - 3Sqrt[Sqrt[#]]] == PrimePi[#] &] (* Alonso del Arte, Jun 27 2011 *)

isA192231(n)=nextprime(floor(n+1-3*n^.25))>n \\ Charles R Greathouse IV, Jun 27 2011

is(n)=for(k=0,sqrtnint(81*n-1,4),if(isprime(n-k), return(0))); 1 \\ Charles R Greathouse IV, Aug 26 2015

a(2) added by Alonso del Arte, Jun 27 2011

A192227 Number of primes in the range (n - 2*sqrt(sqrt(n)), n].

Original entry on oeis.org

0, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 0, 1, 1, 2, 2, 2, 1, 1, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1

Offset: 1

Juri-Stepan Gerasimov, Jun 26 2011

nonn

a(n) is probably positive for all n > 191913030. - Charles R Greathouse IV, Jul 01 2011

Cf. A188817, A189025, A192226.

A192227 := proc(n) local nhi, nlo ; nhi := n ; nlo := floor( n-2*root[4](n)) ; numtheory[pi](nhi)-numtheory[pi](nlo) ; end proc; # R. J. Mathar, Jul 12 2011

Table[PrimePi[n]-PrimePi[n-2*Sqrt[Sqrt[n]]],{n,90}] (* Harvey P. Dale, Feb 24 2018 *)

Conjecturally, a(n) ~ 2*n^(1/4)/log n. - Charles R Greathouse IV, Jul 01 2011

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A189024 Number of primes in the range (n - sqrt(n), n].

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A189026 There appear to be at least n primes in the range (x-sqrt(x), x] for all x >= a(n).

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A189027 There appear to be at least n primes in the range (x-2*sqrt(x), x] for all x >= a(n).

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A192226 Numbers n such that all integers in the interval (n-2*sqrt(sqrt(n)), n] are composite.

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A192231 Numbers k without prime numbers in the range (k-3*sqrt(sqrt(k)), k].

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A192227 Number of primes in the range (n - 2*sqrt(sqrt(n)), n].

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